Inspired by the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, we present exact solutions for a spin-1 chain with Kitaev-like couplings. We consider an expanded Kitaev model with bilinear and biquadratic terms. At an exactly solvable point, the Hamiltonian can be reexpressed as a sum of projection operators. Unlike the AKLT model where projectors act on total spin, we project the component of spin along the bond direction. This leads to exponential ground-state degeneracy, expressed in terms of fractionalized spin-$\frac{1}{2}$ objects. Each ground state can be expressed concisely as a matrix product state. We construct a phase diagram by varying the relative strength of bilinear and biquadratic terms. The fractionalized states provide a qualitative picture for the spin-1 Kitaev model, yielding approximate forms for the ground state and low-lying excitations.

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• Fractionalization
• Frustrated magnetism
• Exact diagonalization
• Exact solutions for many-body systems
• Kitaev model
• Matrix product states